Method for impedance prediction of a voltage source converter under variable operating points

ABSTRACT

A method for predicting impedance of a voltage source converter under variable operating points is disclosed, pertaining to the field of analysis of small signal stability in an electrical power system. When the parameters and topology of the control system are fully unknown, the theoretical expression of impedance cannot be applied to a voltage source converter, and impedance is highly dependent on operating points. A fully decoupled impedance model disclosed is constructed in this disclosure. The fully decoupled impedance model decomposes the impedance into a system parameter vector and an operating point vector, and then identifies parameters through impedance measurement, thereby realizing impedance prediction in a voltage source converter grid-connected system under variable operating points when the parameters and topology of the control system are fully unknown, expanding the scope of application of the impedance method and laying a foundation for using the impedance method to analyze small signal stability in engineering practices.

TECHNICAL FIELD

The present invention pertains to the field of analysis of small signalstability in an electrical power system and more specifically, relatesto a method for impedance prediction of a voltage source converter undervariable operating points.

BACKGROUND ART

Due to the increasing pressure from environmental protection and energyresources, a large amount of solar energy, wind energy and distributedpower supplies are connected to the grid, and the traditional powersystem has gradually been transformed into a power-electronics-basedpower system. As one of the most common devices, voltage sourceconverters (VSC) have been widely used in photovoltaics (PV), wind farmsand high-voltage direct current (HVDC) transmission systems. At present,we are facing various serious stability and oscillation problems causedby power electronic devices, so it is very necessary to carry outaccurate mathematical modeling and analysis of VSC.

At present, there are two major methods for analyzing small signalstability in a power-electronics-based power system. They are thestate-space method in the time domain and the impedance method in thefrequency domain. Generally, the state-space model is feasible only whenthe system structures and parameters are fully known. Similarly, animpedance model of the power electronic devices connected to the gridcan be derived theoretically. However, for the sake of trade secrets, weusually are unable to obtain all the structures and parameters of thepower electronic devices, while in experiments, the device impedance canalso be obtained through injection and sweep frequency measurement ofseries voltage or shunt current. According to these impedance resultsand based on the generalized Nyquist stability criterion, the analysisof small signal stability can be further studied. Therefore, theimpedance method has been widely used under various practical conditionsand has become a dominant method in engineering.

The small signal stability is linearized in a small enough area under acertain operating point and the impedance of the device should beobtained under a specific operating point. Even if there is no changeinside the power electronic device, any external changes will causechanges of the operating point of the device, which in turn change theimpedance of the device. This makes impedance measurement under variableoperating points very time-consuming and tedious, especially when wewant to study the stability boundary of the system or look for faultsunder various operating conditions. Therefore, it is a big challenge tofind an appropriate and fast impedance prediction algorithm when theparameters and structures of the power electronic devices are completelyunknown and the operating points are variable. So far, there is not anyimpedance prediction method for the situation where the parameters andstructures of power electronic devices are completely unknown. This hasdirectly limited the applicability of the impedance method inengineering practices.

SUMMARY OF THE INVENTION

To address the above defects or improvement demands of the prior art,the present invention provides a method for impedance prediction of avoltage source converter under variable operating points, for thepurpose of applying the impedance method in the power electronic devicesof which parameters and structures are completely unknown.

In order to achieve the above object, the present invention provides amethod for impedance prediction of a voltage source converter undervariable operating points, comprising the following steps:

S1. Model small signals of a control system and a filter of a voltagesource converter, respectively, and use small signal impedance toconstruct a fully decoupled impedance model of a voltage sourceconverter grid-connected system, wherein intermediate variables obtainedfrom small signal modeling are all linear functions of operating points,and the completely decoupled impedance model decomposes impedanceinformation into a decoupled form of a coefficient parameter vector andan operating point vector;

S2. Measure impedance information under multiple groups of pre-givenoperating points at a frequency point that needs to be predicted, anduse the measured impedance information to identify the system parametervector in the fully decoupled impedance model;

S3. Substitute the operating point vector that needs to be predictedinto the fully decoupled impedance model to obtain a predicted impedancevalue under the corresponding operating point.

Further, the small signal impedance is:

${\begin{bmatrix}g_{i1} & g_{i2} \\g_{i3} & g_{i4}\end{bmatrix}\begin{bmatrix}{{- \Delta}\; I_{d}^{s}} \\{{- \Delta}\; I_{q}^{s}}\end{bmatrix}} = {\begin{bmatrix}g_{v1} & g_{v2} \\g_{v3} & g_{v4}\end{bmatrix}\begin{bmatrix}{\Delta V_{d}^{s}} \\{\Delta V_{q}^{s}}\end{bmatrix}}$

where ΔI_(d) ^(s), ΔI_(g) ^(s), ΔV_(d) ^(s), ΔV_(q) ^(s) respectivelyrepresent the values of the small signals of the voltage and current atthe grid connection point under the dq coordinate system of the systemand the direction from the device to the grid connection point isspecified as a positive direction of current; and g_(i1), g_(i2),g_(i3), g_(i4), g_(v1), g_(v2), g_(v3), g_(v4) are formed throughcoupling of system parameters, disturbance frequency and operating pointinformation and are in polynomial nonlinear relations with operatingpoints.

Further, the fully decoupled impedance model is:

$\left\{ {\begin{matrix}{Z_{dd} = {x^{T}A_{11}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}} \\{Z_{dq} = {x^{T}A_{12}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}} \\{Z_{qd} = {x^{T}A_{21}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}} \\{Z_{qq} = {x^{T}A_{22}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}}\end{matrix}\left\{ \begin{matrix}{A_{0} = {{b_{1}b_{4}^{T}} - {b_{2}b_{3}^{T}}}} \\{A_{11} = {{a_{1}b_{4}^{T}} - {a_{3}b_{2}^{T}}}} \\{A_{12} = {{a_{2}b_{4}^{T}} - {a_{4}b_{2}^{T}}}} \\{A_{21} = {{a_{3}b_{1}^{T}} - {a_{1}b_{3}^{T}}}} \\{A_{22} = {{a_{4}b_{1}^{T}} - {a_{2}b_{3}^{T}}}}\end{matrix} \right.} \right.$

x represents a combination of polynomial relations of operating points,where the combination comprises constant terms, first-order terms,quadratic terms and multiple terms; a_(k)=[a_(k1) a_(k2) . . .a_(kL)]^(T); b_(k)=[b_(k1) b_(k2) . . . b_(kL)]^(T), k=1, 2, 3, 4, and Lis the length of the operating point vector; and Z_(dd), Z_(dq), Z_(qd),Z_(qq) are four elements of the impedance matrix.

Further, S2 specifically comprises the following steps:

01. Substitute the measured impedance information into the followingformula to obtain a coefficient matrix;

$\left\{ {\begin{matrix}{{M_{1}\rho} = 0} \\{{X^{T}a_{2}} = {M_{2}^{T}\begin{bmatrix}b_{1}^{T} & b_{2}^{T}\end{bmatrix}}^{T}} \\{{X^{T}a_{4}} = {M_{2}^{T}\begin{bmatrix}b_{3}^{T} & b_{4}^{T}\end{bmatrix}}^{T}}\end{matrix}{where}\left\{ \begin{matrix}{m_{1} = \begin{bmatrix}{{- Z_{dd}}x^{T}} & {{- Z_{qd}}x^{T}} & x^{T}\end{bmatrix}} \\{m_{2} = \begin{bmatrix}{Z_{dq}x^{T}} & {Z_{qq}x^{T}}\end{bmatrix}} \\{\rho_{1} = \begin{bmatrix}b_{1}^{T} & b_{2}^{T} & a_{1}^{T}\end{bmatrix}} \\{\rho_{2} = \begin{bmatrix}b_{3}^{T} & b_{4}^{T} & a_{3}^{T}\end{bmatrix}}\end{matrix} \right.} \right.$

M₁=[m₁₁ m₁₁ . . . m_(1N)], and m_(1k) represents the value of m₁ undergroup k of operating points; ρ represents a set of general solutions ofthe equation set, and ρ₁, ρ₂ are solution elements; X=[x₁ x₂ . . .x_(N)], and x_(k) represents the value of x under group k of operatingpoints; M₂=[m₂₁ m₂₁ . . . m_(2N)], and m_(2k) represents the value of m₂under group k of operating points; and N is the number of groups ofpre-given operating points;

02. Solve the linear equation set corresponding to the coefficientmatrix, and increase the length of the preset operating point vector andre-solve the equation set if the equation set has less than two systemsof fundamental solutions, thereby identifying the system parametervector in the fully decoupled impedance model.

Further, the number of groups of pre-given operating points meets aprecondition: it should ensure that selecting different fundamentalsystems of solutions from the set ρ of general solutions to constituteρ₁, ρ₂ is irrelevant to the impedance prediction result.

Further, when the precondition is not met, the number of given operatingpoints is increased and the linear equation set corresponding to thecoefficient matrix is re-solved.

Further, the number of groups of pre-given operating points is 3L−2.

On the whole, compared with the prior art, the above technical solutionsconceived by the present invention have the following beneficialeffects:

When the parameters and topology of the control system are fullyunknown, the theoretical expression of impedance cannot be applied to avoltage source converter, and impedance is highly dependent on operatingpoints, impeding the direct use of the generalized Nyquist criterionfrom judging small signal stability. The present invention constructs afully decoupled impedance model, which decomposes impedance into asystem parameter vector and an operating point vector, and thenidentifies parameters through impedance measurement, thereby realizingimpedance prediction in a voltage source converter grid-connected systemunder variable operating points when the parameters and topology of thecontrol system are fully unknown, and laying a foundation for using theimpedance method to analyze small signal stability in engineeringpractices.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow diagram of a method for impedance prediction of avoltage source converter under variable operating points provided by thepresent invention;

FIG. 2 is a schematic view of a typical voltage source converter (VSC)grid-connected test system provided by an embodiment of the presentinvention;

FIG. 3 is a comparison diagram of an impedance prediction result at apreset operating point provided by an embodiment of the presentinvention and an impedance measurement result.

DETAILED DESCRIPTION

In order to make the object, technical solutions and advantages of thepresent invention clearer, the present invention will be furtherdescribed in detail in conjunction with the drawings and embodiments. Itshould be understood that the specific embodiments described herein areintended to explain the present invention only and not to limit thepresent invention. Further, the technical features involved in theembodiments of the present invention described below can be combinedwith each other as long as they do not conflict with each other.

As shown in FIG. 1 , the present invention discloses a method forimpedance prediction of a voltage source converter under variableoperating points, comprising the following steps:

S1. Establish a fully decoupled impedance model of a voltage sourceconverter (VSC) grid-connected system according to the definition ofsmall signal impedance, and decompose impedance information into adecoupled form of a coefficient parameter vector and an operating pointvector.

Converting three-phase variables to the dq reference system is a verypractical means for converter control modeling. No matter what controltopology and parameters are contained in the VSC grid-connected system,the system always contains the dq reference system (d^(c)-q^(c)) of thecontroller and the dq reference system (d^(s)-q^(s)) of the system, andthe transformation relationship between the two coordinate systems isdetermined by matrix T_(Δθ),

$T_{\Delta\theta} = \begin{bmatrix}{{\cos\;({\Delta\theta})}\;} & {\sin\left( {\Delta\;\theta} \right)} \\{- {\sin\left( {\Delta\;\theta} \right)}} & {\cos({\Delta\theta})}\end{bmatrix}$

where Δθ represents an angle difference between the two coordinatesystems, which generally is caused by a dynamic process, i.e., in asteady state, Δθ=0. The three-phase voltage and current at the VSC gridconnection point are converted to the two dq coordinate systems,respectively, and small signal processing is conducted to obtain

$\begin{bmatrix}{\Delta V_{d}^{c}} \\{\Delta V_{q}^{c}}\end{bmatrix} = {{\begin{bmatrix}{\Delta V_{d}^{s}} \\{\Delta V_{q}^{s}}\end{bmatrix} + {{{\Delta\theta}\begin{bmatrix}V_{q} \\{- V_{d}}\end{bmatrix}}\begin{bmatrix}{\Delta\; I_{d}^{c}} \\{\Delta\; I_{q}^{c}}\end{bmatrix}}} = {{\begin{bmatrix}{\Delta\; I_{d}^{s}} \\{\Delta\; I_{q}^{s}}\end{bmatrix} + {{{\Delta\theta}\begin{bmatrix}I_{q} \\{- I_{d}}\end{bmatrix}}\begin{bmatrix}{\Delta e_{d}^{c}} \\{\Delta e_{q}^{c}}\end{bmatrix}}} = {\begin{bmatrix}{\Delta e_{d}^{s}} \\{\Delta e_{q}^{s}}\end{bmatrix} + {{\Delta\theta}\begin{bmatrix}e_{q} \\{- e_{d}}\end{bmatrix}}}}}$

where V_(d), V_(q), I_(d), I_(q) are the steady values of thethree-phase voltage and current at the VSC grid connection point in thedq coordinate system, and e_(d), e_(q) are the steady values of theinner electric potential of the VSC in the dq coordinate system. ΔV_(d)^(c), ΔV_(q) ^(c), ΔI_(d) ^(c), ΔI_(q) ^(c), Δe_(d) ^(c), Δe_(q) ^(c),ΔV_(d) ^(s), ΔV_(q) ^(s), ΔI_(d) ^(s), ΔI_(q) ^(s), Δe_(d) ^(s), Δe_(q)^(s) respectively correspond to the values of their small signals in thedq reference system of control and the dq reference system of thesystem.

The control system of the VSC is divided into two parts, which aresynchronous links and other control links. The role of the synchronouslinks is to generate a reference angle θ in the control system, and therole of the other control links is to design a control loop. Thereference angle is an angle of the Park transformation from thethree-phase coordinate system to the dq coordinate system.

Currently, there are two main types of synchronous links, namely:network construction type and network follower type. The reference angleθ of the network follower type is generated by a phase-locked loop, andthe reference angle θ of the network construction type is generated bypower imbalance, for example: droop control. Through small signalmodeling of different synchronous links, the following general form canbe obtained:

${\Delta\theta} = {{\left\lbrack {\frac{f_{din}}{f_{did}}\ \frac{f_{qin}}{f_{qid}}} \right\rbrack\begin{bmatrix}{\Delta\; I_{d}^{s}} \\{\Delta\; I_{q}^{s}}\end{bmatrix}} + {\left\lbrack {\frac{f_{dvn}}{f_{dvd}}\ \frac{f_{qvn}}{f_{qvd}}} \right\rbrack\begin{bmatrix}{\Delta V_{d}^{s}} \\{\Delta V_{q}^{s}}\end{bmatrix}}}$

where f_(din), f_(did), f_(qin), f_(qid), f_(dvn), f_(dvd), f_(qvn),f_(qvd) are linear functions of operating point (V_(d), V_(q), I_(d),I_(q)). Generally, V_(q)=0, V_(d)=V_(t), and V_(t) is the amplitude ofthe voltage at the grid connection point.

For the other control links, after the operating point is linearized,all intermediate variables are linear functions of the operating point,i.e.,

$\begin{bmatrix}{\Delta e_{d}^{c}} \\{\Delta e_{q}^{c}}\end{bmatrix} = {{\begin{bmatrix}f_{ic1} & f_{ic2} \\f_{ic3} & f_{ic4}\end{bmatrix}\begin{bmatrix}{\Delta\; I_{d}^{c}} \\{\Delta\; I_{q}^{c}}\end{bmatrix}} + {\begin{bmatrix}f_{is1} & f_{is2} \\f_{is3} & f_{is4}\end{bmatrix}{\quad{\begin{bmatrix}{\Delta\; I_{d}^{s}} \\{\Delta\; I_{q}^{s}}\end{bmatrix} + {\begin{bmatrix}f_{vc1} & f_{vc2} \\f_{vc3} & f_{vc4}\end{bmatrix}\begin{bmatrix}{\Delta V_{d}^{c}} \\{\Delta V_{q}^{c}}\end{bmatrix}} + {\begin{bmatrix}f_{vs1} & f_{vs2} \\f_{vs3} & f_{vs4}\end{bmatrix}\begin{bmatrix}{\Delta V_{d}^{s}} \\{\Delta V_{q}^{s}}\end{bmatrix}}}}}}$

where f_(ic1˜4), f_(is1˜4), f_(vc1˜4), f_(vs1˜4), are all linearfunctions of operating point (V_(t), I_(d), I_(q)). Further, throughsmall signal modeling of the VSC filter, we may get

$\begin{bmatrix}{\Delta e_{d}^{s}} \\{\Delta e_{q}^{s}}\end{bmatrix} = {{\begin{bmatrix}f_{li1} & f_{li2} \\f_{li3} & f_{li4}\end{bmatrix}\begin{bmatrix}{\Delta\; I_{d}^{s}} \\{\Delta\; I_{q}^{s}}\end{bmatrix}} + {\begin{bmatrix}f_{lv1} & f_{lv2} \\f_{lv3} & f_{lv4}\end{bmatrix}\begin{bmatrix}{\Delta V_{d}^{s}} \\{\Delta V_{q}^{s}}\end{bmatrix}}}$

where f_(li1˜4), f_(lv1˜4) are all linear functions of the operatingpoint (V_(t), I_(d), I_(q)). The above formulae are combined and Δe_(d)^(s), Δe_(q) ^(s), Δe_(d) ^(c), Δe_(q) ^(c) are cancelled to obtain:

${\begin{bmatrix}f_{i\; 1} & f_{i\; 2} \\f_{i\; 3} & f_{i\; 4}\end{bmatrix}\begin{bmatrix}{{- \Delta}\; I_{d}^{s}} \\{{- \Delta}\; I_{q}^{s}}\end{bmatrix}} = {{\begin{bmatrix}f_{v1} & f_{v2} \\f_{v3} & f_{v4}\end{bmatrix}\begin{bmatrix}{\Delta V_{d}^{s}} \\{\Delta V_{q}^{s}}\end{bmatrix}} + {{\Delta\theta}\begin{bmatrix}f_{d\;\theta} \\f_{q\theta}\end{bmatrix}}}$ where $\left\{ \begin{matrix}{{f_{i\; 1} = {f_{{li}\; 1} - f_{{ic}\; 1} - f_{{is}\; 1}}},{f_{i\; 2} = {f_{{li}\; 2} - f_{{ic}\; 2} - f_{{is}\; 2}}}} \\{{f_{i\; 3} = {f_{{li}\; 3} - f_{{ic}\; 3} - f_{{is}\; 3}}},{f_{i\; 4} = {f_{{li}\; 4} - f_{{ic}\; 4} - f_{{is}\; 4}}}} \\{{f_{v1} = {f_{{lv}\; 1} - f_{{vc}\; 1} - f_{{vs}\; 1}}},{f_{v2} = {f_{{lv}\; 2} - f_{vc2} - f_{vs2}}}} \\{{f_{v\; 3} = {f_{{lv}\; 3} - f_{{vc}\; 3} - f_{{vs}\; 3}}},{f_{v\; 4} = {f_{{lv}\; 4} - f_{{vc}\; 4} - f_{{vs}\; 4}}}} \\{{f_{d\;\theta} = {e_{q} - {f_{{ic}\; 1}I_{q}} + f_{{ic}\; 2}}},{I_{d} + {f_{vc2}V_{t}}}} \\{{f_{q\theta} = {{- e_{d}} - {f_{{ic}\; 3}I_{q}} + f_{{ic}\; 4}}},{I_{d} + {f_{{vc}\; 4}V_{t}}}}\end{matrix} \right.$

Apparently, f_(i1˜4), f_(v1˜4) are all linear functions of the operatingpoint (V_(t), I_(d), I_(q)). In particular, as e_(d), e_(q) can belinearly expressed with the operating point, f_(dθ), f_(qθ) are alsolinear functions of the operating point.

By substituting the Δθ expression in the synchronous links into theabove formula, we get

${\begin{bmatrix}g_{i1} & g_{i2} \\g_{i\; 3} & g_{i\; 4}\end{bmatrix}\begin{bmatrix}{{- \Delta}\; I_{d}^{s}} \\{{- \Delta}\; I_{q}^{s}}\end{bmatrix}} = {\begin{bmatrix}g_{v1} & g_{v2} \\g_{v3} & g_{v4}\end{bmatrix}\begin{bmatrix}{\Delta V_{d}^{s}} \\{\Delta V_{q}^{s}}\end{bmatrix}}$ where $\left\{ \begin{matrix}{g_{i\; 1} = {\left( {{f_{i\; 1}f_{did}} + {f_{d\;\theta}f_{din}}} \right)f_{qid}f_{dvd}f_{qvd}}} \\{g_{i2} = {\left( {{f_{i\; 2}f_{qid}} + {f_{d\;\theta}f_{qin}}} \right)f_{did}f_{dvd}f_{qvd}}} \\{g_{i3} = {\left( {{f_{i\; 1}f_{did}} + {f_{d\;\theta}f_{din}}} \right)f_{qid}f_{dvd}f_{qvd}}} \\{g_{i4} = {\left( {{f_{i4}f_{qid}} + {f_{q\theta}f_{qin}}} \right)f_{did}f_{dvd}f_{qvd}}} \\{g_{v\; 1} = {\left( {{f_{v\; 1}f_{dvd}} + {f_{d\;\theta}f_{dvn}}} \right)f_{did}f_{qid}f_{qvd}}} \\{g_{v2} = {\left( {{f_{v2}f_{qvd}} + {f_{d\;\theta}f_{qvn}}} \right)f_{did}f_{qid}f_{dvd}}} \\{g_{v3} = {\left( {{f_{v3}f_{dvd}} + {f_{q\theta}f_{dvn}}} \right)f_{did}f_{qid}f_{qvd}}} \\{g_{v4} = {\left( {{f_{v4}f_{qvd}} + {f_{q\theta}f_{qvn}}} \right)f_{did}f_{qid}f_{dvd}}}\end{matrix} \right.$

It can be seen that g_(i1˜4), g_(v1˜4) have polynomial nonlinearrelations with the operating point. The preset operating point vectorcontains the constant terms, first-order terms, quadratic terms andmultiple terms of the operating point, and the length is L. They arewritten in a vector form to obtain:

$\left\{ \begin{matrix}{g_{ik} = {a_{k}^{T}x}} \\{g_{ik} = {b_{k}^{T}x}}\end{matrix} \right.,{k = 1},2,3,4$

where x represents a combination of polynomial relations of operatingpoints, a_(k)=[a_(k1) a_(k2) . . . a_(kL)]^(T); b_(k)=[b_(k1) b_(k2) . .. b_(kL)]^(T); supposing L=10, then:

$\quad\left\{ \begin{matrix}{x = \left\lbrack {1\mspace{14mu} I_{d}\mspace{14mu} I_{q}\mspace{9mu} V_{t}\mspace{20mu} I_{d}^{2}\mspace{20mu} I_{q}^{2}\mspace{14mu} I_{d}\; I_{q}\mspace{20mu} I_{d}V_{t}\mspace{14mu} I_{q}V_{t}\mspace{14mu} V_{t}^{2}} \right\rbrack^{T}} \\{a_{k} = \left\lbrack {a_{k1}\mspace{14mu} a_{k2}\mspace{14mu} a_{k3}\mspace{14mu} a_{k4}\mspace{14mu} a_{k5}\mspace{14mu} a_{k6}\mspace{14mu} a_{k7}\mspace{14mu} a_{k8}\mspace{14mu} a_{k9}\mspace{14mu} a_{k\; 10}} \right\rbrack^{T}} \\{b_{k} = {\left\lbrack {b_{k1}\mspace{14mu} b_{k2}\mspace{14mu} b_{k3}\ b_{k4}\mspace{14mu} b_{k5}\mspace{14mu} b_{k6}\mspace{14mu} b_{k7}\mspace{14mu} b_{k8}\mspace{14mu} b_{k9}\mspace{14mu} b_{k10}} \right\rbrack^{T}}^{\;}}\end{matrix} \right.$

where a_(k), b_(k) are system parameter vectors and a_(k1˜10), b_(k1˜10)are system vector parameters relevant to system parameters anddisturbance frequency only.

The definition of impedance matrix Z may be written into:

$\begin{bmatrix}{\Delta V_{d}^{s}} \\{\Delta V_{q}^{s}}\end{bmatrix} = {\begin{bmatrix}Z_{dd} & Z_{dq} \\Z_{qa} & Z_{qq}\end{bmatrix}\begin{bmatrix}{{- \Delta}\; I_{d}^{s}} \\{{- \Delta}\; I_{q}^{s}}\end{bmatrix}}$

where Z_(dd), Z_(dq), Z_(qd), Z_(qq) are four elements of the impedancematrix, and the expression is as follows:

$\quad\left\{ \begin{matrix}{Z_{dd} = {\left( {{g_{i1}g_{v4}} - {g_{v2}g_{i3}}} \right)\Delta_{gv}^{- 1}}} \\{Z_{dq} = {\left( {{g_{i2}g_{v4}} - {g_{v2}g_{i4}}} \right)\Delta_{gv}^{- 1}}} \\{Z_{qd} = {\left( {{g_{i3}g_{v1}} - {g_{v3}g_{i1}}} \right)\Delta_{gv}^{- 1}}} \\{Z_{qq} = {\left( {{g_{i4}g_{v1}} - {g_{v3}g_{i2}}} \right)\Delta_{gv}^{- 1}}}\end{matrix} \right.$

where Δ_(gv)=g_(v1)g_(v4)-g_(v2)g_(v3).

Thus, the fully decoupled impedance model is:

$\quad\left\{ {\begin{matrix}{Z_{dd} = {x^{T}A_{11}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}} \\{Z_{dq} = {x^{T}A_{12}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}} \\{Z_{qd} = {x^{T}A_{21}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}} \\{Z_{qq} = {x^{T}A_{22}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}}\end{matrix}{where}\left\{ {\begin{matrix}{A_{0} = {{b_{1}b_{4}^{T}} - {b_{2}b_{3}^{T}}}} \\{A_{11} = {{b_{1}b_{4}^{T}} - {a_{3}b_{2}^{T}}}} \\{A_{12} = {{b_{2}b_{4}^{T}} - {a_{4}b_{2}^{T}}}} \\{A_{21} = {{b_{3}b_{1}^{T}} - {a_{1}b_{3}^{T}}}} \\{A_{22} = {{b_{4}b_{1}^{T}} - {a_{2}b_{3}^{T}}}}\end{matrix}.} \right.} \right.$

S2. Measure impedance information under multiple groups of pre-givenoperating points at a frequency point that needs to be predicted andobtain a coefficient matrix through substitution. Solve the linearequation set corresponding to the coefficient matrix, and increase thelength of the preset operating point vector and re-solve the equationset if the equation set has less than two systems of fundamentalsolutions, thereby identifying the system parameter vector in the fullydecoupled impedance model.

According to the above derivation, we can get:

$\quad\left\{ \begin{matrix}{{{g_{i1}\Delta\; I_{d}^{s}} + {g_{i2}\Delta\; I_{q}^{s}} + {g_{v1}\Delta V_{d}^{s}} + {g_{v2}\Delta V_{q}^{s}}} = 0} \\{{{g_{i3}\Delta\; I_{d}^{s}} + {g_{i4}\Delta\; I_{q}^{s}} + {g_{v3}\Delta V_{d}^{s}} + {g_{v4}\Delta V_{q}^{s}}} = 0}\end{matrix} \right.$

According to the definition of impedance, we can get:

$\quad\left\{ \begin{matrix}{{\Delta V_{d}^{s}} = {{{- Z_{dd}}\Delta\; I_{d}^{s}} - {Z_{dq}\Delta\; I_{q}^{s}}}} \\{{\Delta V_{q}^{s}} = {{{- Z_{qd}}\Delta\; I_{d}^{s}} - {Z_{qq}\Delta\; I_{q}^{s}}}}\end{matrix} \right.$

By substitution, we may get:

$\left\{ {\begin{matrix}{{{\sum\limits_{1}{\Delta\; I_{d}^{s}}} + {\sum\limits_{2}{\Delta\; I_{q}^{s}}}} = 0} \\{{{\sum\limits_{3}{\Delta\; I_{d}^{s}}} + {\sum\limits_{4}{\Delta\; I_{q}^{s}}}} = 0}\end{matrix}{where}\left\{ \begin{matrix}{\sum\limits_{1}{= {{{- Z_{dd}}g_{v\; 1}} - {Z_{qd}g_{v\; 2}} + g_{i\; 1}}}} \\{\sum\limits_{2}{= {{{- Z_{dq}}g_{v\; 1}} - {Z_{qq}g_{v\; 2}} + g_{i\; 2}}}} \\{\sum\limits_{3}{= {{{- Z_{dd}}g_{v\; 3}} - {Z_{qd}g_{v\; 4}} + g_{i\; 3}}}} \\{\sum\limits_{4}{= {{{- Z_{dq}}g_{v\; 3}} - {Z_{qq}g_{v\; 4}} + g_{i\; 4}}}}\end{matrix} \right.} \right.$

As the above equation is tenable under any disturbance, ΔI_(d) ^(s),ΔI_(q) ^(s), may be any value, i.e., when and only when:

$\quad\left\{ \begin{matrix}{{\sum\limits_{1}{= 0}},{\sum\limits_{2}{= 0}}} \\{{\sum\limits_{2}{= 0}},{\sum\limits_{3}{= 0}}}\end{matrix} \right.$

by substituting the vector forms of g_(i1˜4), g_(v1˜4), we may get:

$\quad\left\{ \begin{matrix}{{{{- Z_{dd}}b_{1}^{T}x} - {Z_{qd}b_{2}^{T}x} + {a_{1}^{T}x}} = 0} \\{{{{- Z_{dq}}b_{1}^{T}x} - {Z_{qq}b_{2}^{T}x} + {a_{2}^{T}x}} = 0} \\{{{{- Z_{dd}}b_{3}^{T}x} - {Z_{qd}b_{4}^{T}x} + {a_{3}^{T}x}} = 0} \\{{{{- Z_{dq}}b_{3}^{T}x} - {Z_{qq}b_{4}^{T}x} + {a_{4}^{T}x}} = 0}\end{matrix} \right.$

By sorting it into a matrix form, we may get

$\left\{ {\begin{matrix}{{m_{1}^{T}\rho_{1}} = 0} \\{{m_{1}^{T}\rho_{2}} = 0} \\{{x^{T}a_{2}} = {m_{2}^{T}\left\lbrack {b_{1}^{T}\mspace{9mu} b_{2}^{T}} \right\rbrack}^{T}} \\{{x^{T}a_{4}} = {m_{2}^{T}\left\lbrack {b_{3}^{T}\mspace{9mu} b_{4}^{T}} \right\rbrack}^{T}}\end{matrix}{where}\left\{ \begin{matrix}{m_{1} = \left\lbrack {{{- Z_{dd}}x^{T}}\; - {Z_{qd}x^{T}\mspace{14mu} x^{T}}}\  \right\rbrack} \\{m_{2} = \left\lbrack {Z_{dq}x^{T}\mspace{14mu} Z_{qq}x^{T}} \right\rbrack} \\{\rho_{1} = \left\lbrack {b_{1}^{T}\mspace{14mu} b_{2}^{T}\mspace{14mu} a_{1}^{T}} \right\rbrack} \\{\rho_{2} = \left\lbrack {b_{3}^{T}\mspace{20mu} b_{4}^{T}\mspace{20mu} a_{3}^{T}} \right\rbrack}\end{matrix} \right.} \right.$

By substituting the known operating point vector in group 3L−2 and themeasured corresponding impedance information, we get an equation set:

$\quad\left\{ \begin{matrix}{{M_{1}\rho} = 0} \\{{X^{T}a_{2}} = {M_{2}^{T}\left\lbrack {b_{1}^{T}\mspace{14mu} b_{2}^{T}} \right\rbrack}^{T}} \\{{X^{T}a_{4}} = {M_{2}^{T}\left\lbrack {b_{3}^{T}\mspace{20mu} b_{4}^{T}} \right\rbrack}^{T}}\end{matrix} \right.$

where M₁=[m₁₁ m₁₁ . . . m_(1N)], and m_(1k) represents the value of m₁under group k of operating points; X=[x₁ x₂ . . . x_(N)], and x_(k)represents the value of x under group k of operating points; M₂=[m₂₁ m₂₁. . . m_(2N)], and m_(2k) represents the value of m₂ under group k ofoperating points; and N is the number of groups of known operatingpoints, i.e., 3L−2. ρ represents a set of general solutions of theequation set and ρ₁, ρ₂ are solution elements. The equation set M₁ρ=0 issolved at first to obtain ρ₁, ρ₂, and then the obtained b_(1˜4) issubstituted into two remaining inhomogeneous equations to obtain allsystem parameter vectors, i.e., a_(1˜4), b_(1˜4) in the fully decoupledimpedance model.

The preset operating point vector x contains the constant terms,first-order terms and quadratic terms of the operating point and thisassumption may have too many terms, so the equation set M₁ρ=0 may haveexcessive free variables, in other words, this homogenous equation sethas more than two systems of fundamental solutions. Below we willdiscuss whether selecting different ρ₁, ρ₂ from the set ρ of systems offundamental solutions will affect the final impedance result.

Define two vectors,

$\quad\left\{ \begin{matrix}{\alpha = \left\lbrack {b_{1}^{T}\mspace{14mu} b_{2}^{T}\mspace{14mu} a_{1}^{T}\mspace{14mu} a_{2}^{T}} \right\rbrack} \\{\beta = \left\lbrack {b_{3}^{T}\mspace{14mu} b_{4}^{T}\mspace{14mu} a_{3}^{T}\mspace{14mu} a_{4}^{T}} \right\rbrack}\end{matrix} \right.$

For easy writing, define:

$\quad\left\{ \begin{matrix}{\alpha = {{k_{1}\eta_{1}} + {k_{2}\eta_{2}} + \ldots + {k_{n}\eta_{n}}}} \\{\beta = {{k_{1}^{\prime}\eta_{1}} + {k_{2}^{\prime}\eta_{2}} + \ldots + {k_{n}^{\prime}\eta_{n}}}}\end{matrix} \right.$

where η_(1˜n) is obtained by combining a_(1˜4), which are obtained byselecting different systems of fundamental solutions from the set ρ ofsystems of fundamental solutions to constitute ρ₁, ρ₂, and thensubstituting the ρ₁, ρ₂ into two remaining inhomogeneous equation sets;k_(1˜n), k_(1˜n) are any coefficients. It should ensure that selectingdifferent fundamental systems of solutions from the set ρ of systems offundamental solutions to constitute ρ₁, ρ₂ is irrelevant to theimpedance prediction result; in other words, the selection of k_(1˜n),k_(1˜n) does not affect the final impedance prediction.

Definition: E_(j)=[H₁ ^(T) H₂ ^(T) H₃ ^(T) H₄ ^(T)], j=1, 2, 3, 4, wherewhen i=j H_(i)=I_(10×10), and when i≠j, H_(i)=O_(10×10). Therefore, wemay get:

$\quad\left\{ \begin{matrix}{{b_{1} = {E_{1}\alpha}},{b_{2} = {E_{2}\alpha}},{a_{1} = {E_{3}\alpha}},{a_{2} = {E_{4}\alpha}}} \\{{b_{3} = {E_{1}\beta}},{b_{4} = {E_{2}\beta}},{a_{3} = {E_{3}\beta}},{a_{4} = {E_{4}\beta}}}\end{matrix} \right.$

According to the definitions of A₀, A₁₁, A₁₂, A₂₁, A₂₂ in the fullydecoupled impedance model:

$\quad\left\{ \begin{matrix}{A_{0} = {{b_{1}b_{4}^{T}} - {b_{2}b_{3}^{T}}}} \\{A_{11} = {{a_{1}b_{4}^{T}} - {a_{3}b_{2}^{T}}}} \\{A_{12} = {{a_{2}b_{4}^{T}} - {a_{4}b_{2}^{T}}}} \\{A_{21} = {{a_{3}b_{1}^{T}} - {a_{1}b_{3}^{T}}}} \\{A_{22} = {{a_{4}b_{1}^{T}} - {a_{2}b_{3}^{T}}}}\end{matrix} \right.$

we may get:

$\quad\left\{ \begin{matrix}{A_{0} = {\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{\left( {{k_{i}k_{j}^{\prime}} - {k_{j}k_{i}^{\prime}}} \right){E_{1}\left( {{\eta_{i}\eta_{j}^{T}} - {\eta_{j}\eta_{i}^{T}}} \right)}E_{2}^{T}}}}} \\{A_{11} = {\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{\left( {{k_{i}k_{j}^{\prime}} - {k_{j}k_{i}^{\prime}}} \right){E_{3}\left( {{\eta_{i}\eta_{j}^{T}} - {\eta_{j}\eta_{i}^{T}}} \right)}E_{2}^{T}}}}} \\{A_{12} = {\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{\left( {{k_{i}k_{j}^{\prime}} - {k_{j}k_{i}^{\prime}}} \right){E_{4}\left( {{\eta_{i}\eta_{j}^{T}} - {\eta_{j}\eta_{i}^{T}}} \right)}E_{2}^{T}}}}} \\{A_{21} = {\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{\left( {{k_{i}k_{j}^{\prime}} - {k_{j}k_{i}^{\prime}}} \right){E_{1}\left( {{\eta_{i}\eta_{j}^{T}} - {\eta_{j}\eta_{i}^{T}}} \right)}E_{3}^{T}}}}} \\{A_{22} = {\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{\left( {{k_{i}k_{j}^{\prime}} - {k_{j}k_{i}^{\prime}}} \right){E_{1}\left( {{\eta_{i}\eta_{j}^{T}} - {\eta_{j}\eta_{i}^{T}}} \right)}E_{4}^{T}}}}}\end{matrix} \right.$

Take the calculation of impedance Z_(dd) as an example:

${Z_{dd} = {{x^{T}A_{11}{x\left( {x^{T}A_{0}x} \right)}^{- 1}} = \frac{\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{\left( {{k_{i}k_{j}^{\prime}} - {k_{j}k_{i}^{\prime}}} \right){E_{3}\left( {{\eta_{i}\eta_{j}^{T}} - {\eta_{j}\eta_{i}^{T}}} \right)}E_{2}^{T}}}}{\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{n}{\left( {{k_{i}k_{j}^{\prime}} - {k_{j}k_{i}^{\prime}}} \right){E_{1}\left( {{\eta_{i}\eta_{j}^{T}} - {\eta_{j}\eta_{i}^{T}}} \right)}E_{2}^{T}}}}}},$to ensure the result of Z_(dd) is not affected by k_(1˜n), k_(1˜n), aprecondition is when and only when

$\frac{\left( {{k_{i}k_{j}^{\prime}} - {k_{j}k_{i}^{\prime}}} \right){E_{3}\left( {{\eta_{i}\eta_{j}^{T}} - {\eta_{j}\eta_{i}^{T}}} \right)}E_{2}^{T}}{\left( {{k_{i}k_{j}^{\prime}} - {k_{j}k_{i}^{\prime}}} \right){E_{1}\left( {{\eta_{i}\eta_{j}^{T}} - {\eta_{j}\eta_{i}^{T}}} \right)}E_{2}^{T}}$is the same for any i, j. Therefore, after the systems of fundamentalsolutions of M₁ρ=0 are obtained, they should be verified throughsubstitution. In fact, as long as the difference between known operatingpoint vectors is large, this precondition can be met. If it is foundthat this precondition is not met, the number of known operating pointsis increased and re-solving is conducted. In addition, if the equationset M₁ρ=0 has less than two systems of fundamental solutions, it meansthat our assumption of polynomial relations of operating points in apreset operating point vector is not enough. In response, the length ofthe preset operating point vector is increased, for example, a cubicterm (I_(d) ² I_(q), I_(d) ³ . . . ) is added.

Hence, we get an operating point parameter vector a_(1˜4), b_(1˜4) ofthe impedance complete solution model.

S3. Substitute the operating point vector that needs to be predictedinto the model to obtain predicted impedance under any operating point.

Substitute any group of operating point vectors that need to predictimpedance, i.e., polynomial vectors constituted by the voltage andcurrent at the grid connection point x*=[1 I_(d) I_(q) V_(t) I_(d) ²I_(q) ² I_(d)I_(q) I_(d)V_(t) I_(q)V_(t) V_(t) ²]^(T), to obtain thefollowing predicted impedance:

$\left\{ {\begin{matrix}{Z_{dd} = {x^{*T}A_{11}{x^{*}\left( {x^{*T}A_{0}x^{*}} \right)}^{- 1}}} \\{Z_{dq} = {x^{*T}A_{12}{x^{*}\left( {x^{*T}A_{0}x^{*}} \right)}^{- 1}}} \\{Z_{qd} = {x^{*T}A_{21}{x^{*}\left( {x^{*T}A_{0}x^{*}} \right)}^{- 1}}} \\{Z_{qq} = {x^{*T}A_{22}{x^{*}\left( {x^{*T}A_{0}x^{*}} \right)}^{- 1}}}\end{matrix}{where}\left\{ \begin{matrix}{A_{0} = {{b_{1}b_{4}^{T}} - {b_{2}b_{3}^{T}}}} \\{A_{11} = {{a_{1}b_{4}^{T}} - {a_{3}b_{2}^{T}}}} \\{A_{12} = {{a_{2}b_{4}^{T}} - {a_{4}b_{2}^{T}}}} \\{A_{21} = {{a_{3}b_{1}^{T}} - {a_{1}b_{3}^{T}}}} \\{A_{22} = {{a_{4}b_{1}^{T}} - {a_{2}b_{3}^{T}}}}\end{matrix} \right.} \right.$

Compared with the prior art, the present invention has considered thatwhen the parameters and topology of the control system are fullyunknown, the theoretical expression of impedance cannot be applied to avoltage source converter, and impedance is highly dependent on operatingpoints, impeding the direct use of the generalized Nyquist criterionfrom judging small signal stability, and can analyze small signalstability of a voltage source converter grid-connected system withcompletely unknown structures and parameters, expanding the scope ofapplication of the impedance method.

In order for those skilled in the art to better understand the presentinvention, we will describe in detail the use of a calculation methodfor impedance prediction of a voltage source converter under variableoperating points provided by the present invention in impedanceprediction in conjunction with specific embodiments.

Embodiment

A typical VSC grid-connected test system is selected. The system adoptsa phase-locked loop (PLL) 6 as a synchronous link and adopts terminalvoltage and DC voltage outer loop control (DVC&TVC) 14 and AC currentinner loop control (ACC) 12. Through 30 groups of known operating pointvector and impedance information, the system performs impedanceprediction of the operating point vectors that need to be predicted.

The system is a shown in FIG. 2 , where L_(f) represents filterinductance of a filter 4, C represents DC side capacitance, P_(m) ispower input from the DC side, e_(abc) is three-phase inner electricpotential of VSC 8, V_(tabc) is three-phase voltage at the gridconnection point 2, i_(abc) is three-phase current at the gridconnection point 2, V_(d), V_(q), i_(d), i_(q) are the values of thethree-phase voltage and current at the grid connection point 2 under thedq coordinate system, e_(d), e_(q) are the values of e_(abc) under thedq coordinate system, V_(t) is the amplitude of V_(tabc), V_(dc) iscapacitive voltage on the DC side, θ is an angle output of thephase-locked loop, i_(dref), i_(qref) are the reference values of i_(d),i_(q), V_(tref), V_(dcref) are the reference values of V_(t), V_(dc),and k_(p1˜4), k_(i1˜4) are parameters of PI control. The parameters usedin this embodiment are all expressed in a per unit system, the basevalue of power is S_(base)=2MVA, and the base value of voltage isV_(base)=690V. The parameters of the controller PI are as follows:k_(p1)=3.5, k_(i1)=140, k_(p2)=1, k_(i2)=100, k_(p3)=0.3, k_(i3)=160,k_(p4)=50, k_(i4)=2000. The 28 groups of known operating pointinformation are as shown in Table 1.

TABLE 1 No. V_(t)(pu) P(pu) Q(pu) 1 0.9704 0.8219 0.6769 2 0.9657 0.60500.7701 3 0.6392 0.3496 −0.5849 4 0.9824 0.1548 −0.9247 5 0.9786 0.4750−0.5887 6 0.5709 0.2408 −0.4747 7 0.8961 0.8598 −0.2791 8 0.5179 0.4397−0.4495 9 0.8394 0.6360 −0.4082 10 0.6961 0.4563 0.4578 11 0.8530 0.02720.3806 12 0.5231 0.0508 −0.3384 13 0.8474 0.2687 −0.7630 14 0.51720.2269 0.1225 15 0.8828 0.7020 0.5528 16 0.7449 0.3319 −0.2180 17 0.85470.6450 0.3829 18 0.8399 0.5502 0.5667 19 0.5595 0.2788 −0.5145 20 0.67020.3922 0.3702 21 0.8756 0.2234 −0.1004 22 0.8495 0.7569 −0.7804 230.7736 0.1072 0.5426 23 0.6288 0.5286 0.3090 25 0.9071 0.2209 −0.7788 260.6750 0.1327 0.3360 27 0.8080 0.3824 0.2397 28 0.9154 0.5358 −0.0910

The operating point that needs to be predicted is: V_(t)=1.0000,P=0.8000, Q=0.0160. The impedance prediction result and the impedancemeasurement results under this group of operating points are shown inFIG. 3 . It can be seen that the predicted impedance is very consistentwith the measured impedance, proving that the fully decoupled impedancemodel constructed in the present invention is accurate and canaccurately predict the impedance under any operating point.

It can be easily understood by those skilled in the art that theforegoing description includes only preferred embodiments of the presentinvention and is not intended to limit the present invention. All themodifications, identical replacements and improvements within the spiritand principle of the present invention should be in the scope ofprotection of the present invention.

What is claimed is:
 1. A method for predicting impedance of a voltagesource converter (VSC) under changing operating points, comprising: S1.providing a voltage source converter grid-connected system comprising acontrol system and a filter of a voltage source converter (VSC)connected to a grid at a VSC grid connection point to output voltageV_(tabc) and current i_(abc) with small signals at the grid connectionpoint, the step S1 further comprising constructing a decoupled impedancemodel of the voltage source converter grid-connected system under thechanging operating points, wherein constructing the decoupled impedancemodel comprises obtaining variables for a small signal impedance of thesystem, the variables are all linear functions of the changing operatingpoints, and decomposing the small signal impedance into a decoupled formof a system parameter vector and an operating point vector; S2.measuring the voltage V_(tabc) and the current i_(abc) at the VSC gridconnection point to determine impedance values of the VSC under multiplegroups of pre-given operating points at a frequency point that needs tobe predicted, and using the determined impedance values to identify thesystem parameter vector in the decoupled impedance model; and S3. usingthe decoupled impedance model with the identified system parametervector, substituting the operating point vector for an operating pointthat needs to be predicted into the decoupled impedance model to obtaina predicted impedance value under the corresponding operating point. 2.The method of predicting impedance of a voltage source converter underchanging operating points according to claim 1, wherein obtainingvariables for a small signal impedance of the system comprisesrepresenting the small signal impedance as: ${\begin{bmatrix}g_{I1} & g_{I2} \\g_{I3} & g_{I4}\end{bmatrix}\begin{bmatrix}{{- \Delta}I_{d}^{s}} \\{\Delta I_{q}^{s}}\end{bmatrix}} = {\begin{bmatrix}g_{v1} & g_{v2} \\g_{v3} & g_{v4}\end{bmatrix}\begin{bmatrix}{\Delta V_{d}^{s}} \\{\Delta V_{q}^{s}}\end{bmatrix}}$ where ΔI_(d) ^(s), ΔI_(g) ^(s), ΔV_(d) ^(s), ΔV_(q) ^(s)respectively represent the values of the small signals of the voltageand current at the grid connection point under a dq coordinate systemand the direction from the voltage source converter to the gridconnection point is specified as a positive direction of current; andg_(i1), g_(i2), g_(i3), g_(i4), g_(v1), g_(v2), g_(v3), g_(v4) areformed through coupling of system parameters, disturbance frequency andoperating point information and are in polynomial nonlinear relationswith operating points.
 3. The method for predicting impedance of avoltage source converter under changing operating points according toclaim 2, wherein constructing the decoupled impedance model furthercomprises representing the decoupled impedance model as:$\quad\left\{ {\begin{matrix}{Z_{dd} = {x^{T}A_{11}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}} \\{Z_{dq} = {x^{T}A_{12}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}} \\{Z_{qd} = {x^{T}A_{21}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}} \\{Z_{qq} = {x^{T}A_{22}{x\left( {x^{T}A_{0}x} \right)}^{- 1}}}\end{matrix}\left\{ \begin{matrix}{A_{0} = {{b_{1}b_{4}^{T}} - {b_{2}b_{3}^{T}}}} \\{A_{11} = {{a_{1}b_{4}^{T}} - {a_{3}b_{2}^{T}}}} \\{A_{12} = {{a_{2}b_{4}^{T}} - {a_{4}b_{2}^{T}}}} \\{A_{21} = {{a_{3}b_{1}^{T}} - {a_{1}b_{3}^{T}}}} \\{A_{22} = {{a_{4}b_{1}^{T}} - {a_{2}b_{3}^{T}}}}\end{matrix} \right.} \right.$ x represents a combination of polynomialrelations of operating points, where the combination comprises constantterms, first-order terms, quadratic terms and multiple terms;a_(k)=[a_(k1) a_(k2) . . . a_(kL)]^(T); b_(k)=[b_(k1) b_(k2) . . .b_(kL)]^(T), k=1, 2, 3, 4, and L is the length of the operating pointvector; and Z_(dd), Z_(dq), Z_(qd), Z_(qq) are four elements of theimpedance matrix.
 4. The method for predicting impedance of a voltagesource converter under changing operating points according to claim 3,wherein S2 comprises the following steps:
 01. Substituting thedetermined impedance values into the following formula to obtain acoefficient matrix; $\quad{\left\{ \begin{matrix}{{M_{1}\rho} = 0} \\{{X^{T}a_{2}} = {M_{2}^{T}\left\lbrack {b_{1}^{T}\mspace{14mu} b_{2}^{T}} \right\rbrack}^{T}} \\{{X^{T}a_{4}} = {M_{2}^{T}\left\lbrack {b_{3}^{T}\mspace{20mu} b_{4}^{T}} \right\rbrack}^{T}}\end{matrix} \right.{where}\left\{ \begin{matrix}{m_{1} = \left\lbrack {{{- Z_{dd}}x^{T}}\; - {Z_{qd}x^{T}\mspace{14mu} x^{T}}}\  \right\rbrack} \\{m_{2} = \left\lbrack {Z_{dq}x^{T}\mspace{14mu} Z_{qq}x^{T}} \right\rbrack} \\{\rho_{1} = \left\lbrack {b_{1}^{T}\mspace{14mu} b_{2}^{T}\mspace{14mu} a_{1}^{T}} \right\rbrack} \\{\rho_{2} = \left\lbrack {b_{3}^{T}\mspace{20mu} b_{4}^{T}\mspace{20mu} a_{3}^{T}} \right\rbrack}\end{matrix} \right.}$ M₁=[m₁₁ m₁₁ . . . m_(1N)], and m_(1k) representsthe value of m₁ under group k of operating points; ρ represents a set ofgeneral solutions of the equation set, and ρ₁, ρ₂ are solution elements;X=[x₁ x₂ . . . x_(N)], and x_(k) represents the value of x under group kof operating points; M₂=[m₂₁ m₂₁ . . . m_(2N)], and m_(2k) representsthe value of m₂ under group k of operating points; and N is the numberof groups of pre-given operating points; and
 02. Solving the linearequation set corresponding to the coefficient matrix, and increasing thelength of the preset operating point vector and re-solve the equationset if the equation set has less than two systems of fundamentalsolutions, thereby identifying the system parameter vector in thedecoupled impedance model.
 5. The method for predicting impedance of avoltage source converter under changing operating points according toclaim 4, wherein the number of groups of pre-given operating pointsmeets a precondition: it should ensure that selecting differentfundamental systems of solutions from the set ρ of general solutions toconstitute ρ₁, ρ₂ is irrelevant to the impedance prediction result. 6.The method for predicting impedance of a voltage source converter underchanging operating points according to claim 4, wherein when theprecondition is not met, the number of given operating points isincreased and the linear equation set corresponding to the coefficientmatrix is re-solved.
 7. The method for predicting impedance of a voltagesource converter under changing operating points according to claim 5,wherein the number of groups of pre-given operating points is 3L−2.